Math education PhD student | 2nd+ε year | she/her
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Additionally, a discrete mathematician, back-out-of-retirement analyst, math art enthusiast, and proponent of the creative and human elements of math
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I started a math PhD in 2020, mastered out, and (insanely) decided to try it again. It's going rather well this time :)
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My original posts are tagged "seven stories." They're mostly silly little stories related to math or grad school
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I was doing board work in my office with a friend yesterday and at some point just sorta paused. And was like,
Why are there so many green markers? <my officemate> is pretty much responsible for the markers now since I don't work on board often, so it makes sense that there's only one red* that I grabbed this week but there's like, infinite green ones**.
* he's quite red-green colorblind
** there's six, in contrast to the two blue and two black. And now one black cause we threw one out.
~here we see the triangle in its natural habitat, exploring every point on its unit circle in a regular pattern. it smoothly shifts from vertical to horizontal movement guided by the pythagorean theorem as it slowly rotates counterclockwise, producing sine and cosine as it traces this path.
it is in places such as this that the interoperability of mathematics can be seen so clearly, sine, pythagoras, hypotenuse, circle. all acting as one connected system.
(How my department does comphrensive exams if you want actual context.)
Apparently, there's precedence for students "correcting" their written comp after the initial go. I learned this today from my friend who's taking the same comp as me next week (while we discussed how not ready we are).
However, because we're math education students, our written comp isn't related to our research area. It covers measure and spectral theory; topics neither of our advisors have ever studied (we have the same co-advisors). The professor writing and grading the exam is just doing it as a favor to all of us so it'd be a bit much to ask him to grade more things. (Compared to if your advisor does it, in which case it's just part of their work as an advisor.)
Also, our advisors are proctoring the exam, but since they don't know the material and the exam writer will be in China at the time, another professor has been brought in to be on call in case we have questions. This prof is the singular other prof in the department trained in the same research area (as the writer) but does math edu research now.
He's actually the third prof on the project we've been working on this summer so our advisors threw please help? onto our research meeting agenda a week or two ago. We all said we very much appriciated it.
During the same meeting, they were like. Do you want us to stop the clock if you're waiting for a response to a question?
And we were like. Nah. With how long it is, either we have a different problem to work on or it's time for a break.
(The length of quals and written comps here are genuinely insanely and unreasonably long. Like, it's pretty stupid imo. But fine! I'll get hazed!)
During yesterday's meeting, I was emailing back and forth with a student to schedule an interview (as part of the research project) and at one point asked the room, "seems reasonable for me to say preferably Wed or Fri but Tue also works? Thu definitely doesn't work though because comps"
I think everyone said something agreeing that doing an interview on the same day as comps was an absolute no go.
Then me: "yeah. I was planning on Thu being just comps. Nothing else."
Other student: "maybe ice cream? Me and <her boyfriend, also a math grad student> were talking about getting ice cream afterwards."
There was agreement that ice cream after comps was rather reasonable and we said we'd discuss plans later.
i must not go to grad school grad school is the soul-killer grad school is the little-death that brings total obliteration i will face my desire for praise and intellectual stimulation i will permit it to pass over me and through me and when it has gone past i will turn my inner eye to see its path where it has gone there will be nothing
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(How my department does comphrensive exams if you want actual context.)
Apparently, there's precedence for students "correcting" their written comp after the initial go. I learned this today from my friend who's taking the same comp as me next week (while we discussed how not ready we are).
However, because we're math education students, our written comp isn't related to our research area. It covers measure and spectral theory; topics neither of our advisors have ever studied (we have the same co-advisors). The professor writing and grading the exam is just doing it as a favor to all of us so it'd be a bit much to ask him to grade more things. (Compared to if your advisor does it, in which case it's just part of their work as an advisor.)
Also, our advisors are proctoring the exam, but since they don't know the material and the exam writer will be in China at the time, another professor has been brought in to be on call in case we have questions. This prof is the singular other prof in the department trained in the same research area (as the writer) but does math edu research now.
He's actually the third prof on the project we've been working on this summer so our advisors threw please help? onto our research meeting agenda a week or two ago. We all said we very much appriciated it.
(How my department does comphrensive exams if you want actual context.)
Apparently, there's precedence for students "correcting" their written comp after the initial go. I learned this today from my friend who's taking the same comp as me next week (while we discussed how not ready we are).
However, because we're math education students, our written comp isn't related to our research area. It covers measure and spectral theory; topics neither of our advisors have ever studied (we have the same co-advisors). The professor writing and grading the exam is just doing it as a favor to all of us so it'd be a bit much to ask him to grade more things. (Compared to if your advisor does it, in which case it's just part of their work as an advisor.)
This is an explaination of the most common formats for comprehensive exams in my department so I can link this in posts where I ramble about them.
Comprehensive exams are taken after you've chosen your area of research and advisor. Actual requirements are very loose so this is just what's most common.
Most students take a written portion and an oral portion, both covering topics closely related to their research.
What math education students do can better be described as a math portion and an education portion. The math portion is the same as the written portion except we can pick any area of math. The education portion is a written thesis proposal.
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Sometimes, you're mixing up more flour[1] so that you can bake muffins and then you spend over an hour...
converting the original recipe from a volume ratio to mass via weighing the ingredients,
figuring out an appropriate multiplier to fill 2/3-3/4ths of your container[2],
actually mixing the flour,
realizing your multiplier is extremely wrong (ended up doubling it),
mixing more flour,
making a little label for the container (a sticky note with packing tape serving as lamination)
typing up a silly tumblr post about the experience,
finding the original webpage to link it in the tumblr post,
realizing the recipe actually has grams already but the ratios are radically different than what you got,
reweighing the ingredients and recalculating the amounts (but at least you know you want about cups total this time),
making a new label, and
actually finishing the tumblr post.
No, I have not started making the muffins themselves.
[1] I can't have gluten or potatos and have only found ONE gf all purpose flour that doesn't contain potato starch within a managable price range. And that one flour isn't sold anywhere in this region (or the last area I lived in). Thus I make this blend myself.
[2] A large plastic container I bought years ago. I only partially fill it up so that I can just shake it to mix instead of need to use a mixing bowl and whisk.
Links to the recipes (with even more commentary) under the cut.
The original flour: https://www.whattheforkfoodblog.com/2016/01/14/nightshade-free-gluten-free-flour-mix/
By volume:
1 cup white rice flour
1 cup brown rice flour
2/3 cup arrowroot starch
1/3 cup tapioca starch
By mass from my measurements (you can look at the webpage for theirs) to make about 3 cups total:
140 g white rice flour
140 g brown rice flour
80 g arrowroot starch
40 g tapioca starch
Bob's red mill works well for all of the ingredients, but I hop between various brands depending on what's available or on sale. The rice flours needed here are *not* the kind of extremely fine rice flours you often find in asian grocery stores though. (Probably use bob's red mill as a reference even if you end up buying something else.)
I find it to be a very forgiving blend (thankfully, given my conversion/calculation nightmare). From what I've seen, the important thing is to keep the rice flour to starches ratio fairly close, though even that isn't super sensative.
I've done all brown rice flour and all white rice flour and many ratios in between. I've also done many different ratios of arrowroot and tapioca. I think I've even replaced some of the starches with corn starch (not a very large porportion though) because I didn't have quite enough.
(They're indeed correct that the rice flours don't have the same density. However, I found that the difference was only about 20g/cup so I just averaged them.)
Also the muffin recipe: https://meaningfuleats.com/the-best-gluten-free-zucchini-muffins/#wprm-recipe-container-28797
I made it once about two weeks ago and they were great. I also added (about) 1/2 tsp of xantham gum (also chocolate chips and walnuts). Though, tbh, I honestly don't know if xantham gum makes much of a difference in most recipes. I should probably do some tests at some point but not today.
I'm gonna try it with chia eggs this time though. I've found this to be an extremely safe sub for lighter baked goods like cakes or muffins (and it often works great in other baked goods too).
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A flat surface is like a halfway point between a Chess board and Manhattan
(the equations only work on my blog)
Distance on a flat plane is measured by the formula below where Δx and Δy represent the differences between the x values and y values of two points that has a distance being measured:
$$\sqrt{ \lvert \Delta x \rvert^2 + \lvert \Delta y \rvert^2 }$$
If one were to put a point down on the Cartesian plane and highlight all of the points that are 1 unit away from said point, the point would have a circle drawn around the point:
The distance formula is the type of measurement that exists in virtually every context under the Sun. However, this type of measurement doesn't work if you can only move vertically and horizontally:
(image courtesy of Wikipedia)
The diagonal path in green is the shortest path that can be taken between the two points in black. However, if diagonal paths are forbidden, then the best one can do is to go 6 vertical steps and 6 horizontal steps with each of the remaining three paths in red, blue, and yellow representing a shortest path to the two points.
This type of measurement where diagonal movement is disallowed and only orthogonal movement is counted is called the Manhattan distance. The Manhattan cannot exist in a regular Euclidean plane because reality does not have set grid lines unless you find yourself in Manhattan.
If one were to draw a point on a Cartesian plane and the x axis and y axis are taken as the grids of the Manhattan distance, the set of all point that are one unit away form a rhombus.
The points of the rhombus can be arrived at by taking a certain number of steps in the orthogonal directions to reach each of the points. This distance can be given an equation that matches that of the flat plane where Δx and Δy again represent the differences between the x values and y values with the only difference being that the differences are raised to the power of 1 instead of 2 and the sum is in a radical of 1 instead of 2.
$$\sqrt[1]{ \lvert \Delta x \rvert^1 + \lvert \Delta y \rvert^1 } = \Delta x + \Delta y$$
This formula above is identical to the distance formula for the normal flat plane with the only difference being that the 2s of the equations are replaced with 1s. Both formulas can be seen as generalizations of the formula below, where p equals 1 in the Manhattan distance, and p equals 2 for normal distances:
$$\sqrt[p]{ \lvert \Delta x \rvert^p + \lvert \Delta y \rvert^p }$$
This equation means that any value p will form a measurement that measures a distance like a weird looking circle described by the equation x^p+y^p=1.
This is a special case of a superellipse where the semimajor axis and semiminor axis are equal and both are equal to 1 unit.
If p were 4, then the measurement would be based on the squircle:
As p approaches ∞, the unit shape looks more and more like a square:
(image courtesy of Wikipedia)
The limit of this measurement is thus treated as if the defining shape was a square:
In this case, a diagonal movement is treated the same as a movement in the orthogonal directions which is the same type of measure used in chess (this type of measurement is named after Pafnuty Lvovich Chebyshev):
(image courtesy of Wikipedia)
A diagonal movement in the Manhattan distance is 2 units, root 2 on the Euclidean plane, and 1 unit on a chess board. In a way the Euclidean distance is in the middle of a chess board and Manhattan (logarithmically).
D&D 5e uses the chessboard metric for determining distance.
I believe 4e also does while 3.5e uses something that isn't a metric*?? It breaks triangle inequality. Don't remember what the older editions use off the top of my head I'm afraid.
*Formal definition of a metric below the cut but written for a lay person with limited notation.
A metric is a function that takes in two values and outputs a real number. This function must satisfy three conditions:
Positive definite: the distance between any two distinct points is positive and the distance between a point and itself is 0.
Symmetric: the distance between points p and q is equal to the distance between points q and p.
Triangle inequality: the distance between points p and q + distance between points q and r is greater than or equal to the distance between points p and r.
I think the last one benefits a lot from actual mathematical notation: using d(p, q) to mean the distance between points p and q, triangle inequality says d(p, q) + d(q, r) ≥ d(p, r).
Absolute value is a metric and we often write |p-q| to mean d(p, q) even when we're actually using a different metric. All the distances described in the OP are also metrics (I think. Unless I missed something).
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